# Wavefunction of an electron after a slit

To properly understand the double slit experiment with electrons, you need to know what is the wavefunction of the electron after it passes through a slit.

This wavefunction cannot be a plane wave. Indeed for a single slit experiment we would have a uniform impact on the screen which is not what is observed (we should observe something like diffraction through single slit for light).

Thus: what is the wave function of an electron after it has passed through a slit, given that it was a plane wave before for example?

• The transverse momentum components are now the Fourier transform of the slit. Oct 7, 2019 at 20:39
• Are you trying to explain the bright and dark areas of the diffraction pattern? Oct 7, 2019 at 23:20

## 1 Answer

To a reasonable approximation it is a cylindrical wave convolved with the single slit amplitude function in the angular direction. The axis of the cylinder is lies at the center of the slit. (Here I have assumed that the slit is much longer than it is wide and that we don't care about the behavior in the long direction.)

It will keep it's frequency and wave-number (or wave-length if $$\lambda$$ makes you happier than $$k$$), but the direction of the wave-vector will, of course, vary in space.

• To be sure to understand, it is a wave like $\frac{e^{i \vec{k}.\vec{r}}}{\sqrt{r}}$, but $\vec{k}$ being in a plane to make it cylindric ? If not could you write your reasonable approximation solution ? Oct 7, 2019 at 20:16
• Using $\vec{r} = (\rho,\phi,z)$ for coordinates we have wave character given by $\psi \propto \exp \left[ i (k\hat{\rho}) \cdot \vec{r} \right] = \exp ( i k \rho )$. I think your distance dependence is right, but it still remains to stick in the angular amplitude dependence from the slit width (though you can ignore that at first if the slit separation is large compared to their width). Oct 7, 2019 at 20:48
• @dmckee Are these wave functions just a spread of the intensity or do they have a quantum component that would explain the diffraction pattern? Oct 7, 2019 at 23:20
• @PhysicsDave Diffraction patterns aren't explained by quantum, they are explained by waves. You can diffract sound and ripples on a pond and other macroscopc classical waves just fine. The thing that makes two-slit style experiments with differentiated photons or particle hits intersting from a quantum point of view is that you register individual, point-like hits (exhibiting a particle-like behavior) in a pattern that corresponds to diffraction (wavelike behavior) in the same experiment. The math is for waves of probability amplitude. Oct 7, 2019 at 23:29
• @dmckee Good. But I wonder if the author thinks of his "plane wave" (the wave function) as a typical E field wave and based on your response does not know it needs to be a probability wave ... or are they similar enough? Oct 8, 2019 at 2:40